The Diaconis-Freedman chain in a multi-dimensional simplex

In this talk, we would like to consider the asymptotic behavior of a Diaconis-Freedman chain in a multi-dimensional simplex. We will give out some conditions under which we have a unique invariant (stationary) probability measure. In some special cases, we also give out explicit formulas of invariant probability density. Moreover, we completely classify all behaviors of this chain in dimensional two. The method used is based on the iterated function systems. This is the joint work with Juergen Jost and Marc Peigne.